From Rump in a 1983 paper

Percentage Accurate: 18.8% → 100.0%

Time: 7.3s

Alternatives: 4

Speedup: 9.3×

Specification

\[x = 10864 \land y = 18817\]

Math

\[\begin{array}{l} \\ \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* 9.0 (pow x 4.0)) (pow y 4.0)) (* 2.0 (* y y))))

C

double code(double x, double y) {
	return ((9.0 * pow(x, 4.0)) - pow(y, 4.0)) + (2.0 * (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((9.0d0 * (x ** 4.0d0)) - (y ** 4.0d0)) + (2.0d0 * (y * y))
end function

Java

public static double code(double x, double y) {
	return ((9.0 * Math.pow(x, 4.0)) - Math.pow(y, 4.0)) + (2.0 * (y * y));
}

Python

def code(x, y):
	return ((9.0 * math.pow(x, 4.0)) - math.pow(y, 4.0)) + (2.0 * (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(9.0 * (x ^ 4.0)) - (y ^ 4.0)) + Float64(2.0 * Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((9.0 * (x ^ 4.0)) - (y ^ 4.0)) + (2.0 * (y * y));
end

Wolfram

code[x_, y_] := N[(N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 18.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* 9.0 (pow x 4.0)) (pow y 4.0)) (* 2.0 (* y y))))

C

double code(double x, double y) {
	return ((9.0 * pow(x, 4.0)) - pow(y, 4.0)) + (2.0 * (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((9.0d0 * (x ** 4.0d0)) - (y ** 4.0d0)) + (2.0d0 * (y * y))
end function

Java

public static double code(double x, double y) {
	return ((9.0 * Math.pow(x, 4.0)) - Math.pow(y, 4.0)) + (2.0 * (y * y));
}

Python

def code(x, y):
	return ((9.0 * math.pow(x, 4.0)) - math.pow(y, 4.0)) + (2.0 * (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(9.0 * (x ^ 4.0)) - (y ^ 4.0)) + Float64(2.0 * Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((9.0 * (x ^ 4.0)) - (y ^ 4.0)) + (2.0 * (y * y));
end

Wolfram

code[x_, y_] := N[(N[(N[(9.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2 - y \cdot y, y \cdot y, \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 9\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma (- 2.0 (* y y)) (* y y) (* (* x x) (* x (* x 9.0)))))

C

double code(double x, double y) {
	return fma((2.0 - (y * y)), (y * y), ((x * x) * (x * (x * 9.0))));
}

Julia

function code(x, y)
	return fma(Float64(2.0 - Float64(y * y)), Float64(y * y), Float64(Float64(x * x) * Float64(x * Float64(x * 9.0))))
end

Wolfram

code[x_, y_] := N[(N[(2.0 - N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\left(9 \cdot {x}^{4} + \left(\mathsf{neg}\left({y}^{4}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left({y}^{4}\right)\right) + 9 \cdot {x}^{4}\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left({y}^{\left(2 + 2\right)}\right)\right) + 9 \cdot {x}^{4}\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   4. pow-prod-up N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left({y}^{2} \cdot {y}^{2}\right)\right) + 9 \cdot {x}^{4}\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   5. pow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(\left(y \cdot y\right) \cdot {y}^{2}\right)\right) + 9 \cdot {x}^{4}\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   6. pow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right) + 9 \cdot {x}^{4}\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot y\right)\right) \cdot \left(y \cdot y\right) + 9 \cdot {x}^{4}\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   8. fma-define N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(y \cdot y\right), y \cdot y, 9 \cdot {x}^{4}\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   9. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\left(\mathsf{neg}\left(y \cdot y\right)\right), \left(y \cdot y\right), \left(9 \cdot {x}^{4}\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\left(0 - y \cdot y\right), \left(y \cdot y\right), \left(9 \cdot {x}^{4}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot y\right)\right), \left(y \cdot y\right), \left(9 \cdot {x}^{4}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot y\right), \left(9 \cdot {x}^{4}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left(9 \cdot {x}^{4}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left({x}^{4} \cdot 9\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   15. sqr-pow N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left(\left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot 9\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left({x}^{\left(\frac{4}{2}\right)} \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot 9\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left({x}^{2} \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot 9\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   18. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left(\left(x \cdot x\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot 9\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   19. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left(x \cdot \left(x \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot 9\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   20. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot 9\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   21. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(9 \cdot {x}^{\left(\frac{4}{2}\right)}\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   22. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(9 \cdot {x}^{\left(\frac{4}{2}\right)}\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   23. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(9, \left({x}^{\left(\frac{4}{2}\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   24. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(9, \left({x}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   25. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(9, \left(x \cdot x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   26. *-lowering-*.f64 100.0%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(9, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0 - y \cdot y, y \cdot y, x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right)} + 2 \cdot \left(y \cdot y\right) \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right) + \left(0 - y \cdot y\right) \cdot \left(y \cdot y\right)\right) + \color{blue}{2} \cdot \left(y \cdot y\right) \]

   2. sub0-neg N/A

      \[\leadsto \left(x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(y \cdot y\right)\right) \cdot \left(y \cdot y\right)\right) + 2 \cdot \left(y \cdot y\right) \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \left(x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right) - \left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) + \color{blue}{2} \cdot \left(y \cdot y\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) + 2 \cdot \left(y \cdot y\right) \]

   5. +-commutative N/A

      \[\leadsto 2 \cdot \left(y \cdot y\right) + \color{blue}{\left(x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)} \]

   6. sub-neg N/A

      \[\leadsto 2 \cdot \left(y \cdot y\right) + \left(x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)}\right) \]

   7. +-commutative N/A

      \[\leadsto 2 \cdot \left(y \cdot y\right) + \left(\left(\mathsf{neg}\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right) + \color{blue}{x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)}\right) \]

   8. associate-+r+ N/A

      \[\leadsto \left(2 \cdot \left(y \cdot y\right) + \left(\mathsf{neg}\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right) + \color{blue}{x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 - y \cdot y, y \cdot y, \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 9\right)\right)\right)} \]
7. Add Preprocessing

Alternative 2: 18.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot \left(y \cdot y\right)\right)\\ t_1 := \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 9\right)\right)\\ t_2 := t\_1 - t\_0\\ \frac{t\_0 \cdot 4 - t\_2 \cdot t\_2}{y \cdot \left(2 \cdot y\right) + \left(t\_0 - t\_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y (* y y))))
        (t_1 (* (* x x) (* x (* x 9.0))))
        (t_2 (- t_1 t_0)))
   (/ (- (* t_0 4.0) (* t_2 t_2)) (+ (* y (* 2.0 y)) (- t_0 t_1)))))

C

double code(double x, double y) {
	double t_0 = y * (y * (y * y));
	double t_1 = (x * x) * (x * (x * 9.0));
	double t_2 = t_1 - t_0;
	return ((t_0 * 4.0) - (t_2 * t_2)) / ((y * (2.0 * y)) + (t_0 - t_1));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = y * (y * (y * y))
    t_1 = (x * x) * (x * (x * 9.0d0))
    t_2 = t_1 - t_0
    code = ((t_0 * 4.0d0) - (t_2 * t_2)) / ((y * (2.0d0 * y)) + (t_0 - t_1))
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * (y * y));
	double t_1 = (x * x) * (x * (x * 9.0));
	double t_2 = t_1 - t_0;
	return ((t_0 * 4.0) - (t_2 * t_2)) / ((y * (2.0 * y)) + (t_0 - t_1));
}

Python

def code(x, y):
	t_0 = y * (y * (y * y))
	t_1 = (x * x) * (x * (x * 9.0))
	t_2 = t_1 - t_0
	return ((t_0 * 4.0) - (t_2 * t_2)) / ((y * (2.0 * y)) + (t_0 - t_1))

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * Float64(y * y)))
	t_1 = Float64(Float64(x * x) * Float64(x * Float64(x * 9.0)))
	t_2 = Float64(t_1 - t_0)
	return Float64(Float64(Float64(t_0 * 4.0) - Float64(t_2 * t_2)) / Float64(Float64(y * Float64(2.0 * y)) + Float64(t_0 - t_1)))
end

MATLAB

function tmp = code(x, y)
	t_0 = y * (y * (y * y));
	t_1 = (x * x) * (x * (x * 9.0));
	t_2 = t_1 - t_0;
	tmp = ((t_0 * 4.0) - (t_2 * t_2)) / ((y * (2.0 * y)) + (t_0 - t_1));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - t$95$0), $MachinePrecision]}, N[(N[(N[(t$95$0 * 4.0), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(y * N[(2.0 * y), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\left(9 \cdot {x}^{4} + \left(\mathsf{neg}\left({y}^{4}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left({y}^{4}\right)\right) + 9 \cdot {x}^{4}\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left({y}^{\left(2 + 2\right)}\right)\right) + 9 \cdot {x}^{4}\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   4. pow-prod-up N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left({y}^{2} \cdot {y}^{2}\right)\right) + 9 \cdot {x}^{4}\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   5. pow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(\left(y \cdot y\right) \cdot {y}^{2}\right)\right) + 9 \cdot {x}^{4}\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   6. pow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right) + 9 \cdot {x}^{4}\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot y\right)\right) \cdot \left(y \cdot y\right) + 9 \cdot {x}^{4}\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   8. fma-define N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(y \cdot y\right), y \cdot y, 9 \cdot {x}^{4}\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   9. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\left(\mathsf{neg}\left(y \cdot y\right)\right), \left(y \cdot y\right), \left(9 \cdot {x}^{4}\right)\right), \mathsf{*.f64}\left(\color{blue}{2}, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\left(0 - y \cdot y\right), \left(y \cdot y\right), \left(9 \cdot {x}^{4}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot y\right)\right), \left(y \cdot y\right), \left(9 \cdot {x}^{4}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \left(y \cdot y\right), \left(9 \cdot {x}^{4}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left(9 \cdot {x}^{4}\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left({x}^{4} \cdot 9\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   15. sqr-pow N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left(\left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \cdot 9\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left({x}^{\left(\frac{4}{2}\right)} \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot 9\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left({x}^{2} \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot 9\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   18. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left(\left(x \cdot x\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot 9\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   19. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \left(x \cdot \left(x \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot 9\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   20. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot 9\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   21. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(9 \cdot {x}^{\left(\frac{4}{2}\right)}\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   22. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(9 \cdot {x}^{\left(\frac{4}{2}\right)}\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   23. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(9, \left({x}^{\left(\frac{4}{2}\right)}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   24. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(9, \left({x}^{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   25. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(9, \left(x \cdot x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

   26. *-lowering-*.f64 100.0%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(9, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(y, y\right)\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0 - y \cdot y, y \cdot y, x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right)} + 2 \cdot \left(y \cdot y\right) \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto 2 \cdot \left(y \cdot y\right) + \color{blue}{\left(\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right) + x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right)} \]

   2. flip-+ N/A

      \[\leadsto \frac{\left(2 \cdot \left(y \cdot y\right)\right) \cdot \left(2 \cdot \left(y \cdot y\right)\right) - \left(\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right) + x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right) + x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right)}{\color{blue}{2 \cdot \left(y \cdot y\right) - \left(\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right) + x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left(2 \cdot \left(y \cdot y\right)\right) \cdot \left(2 \cdot \left(y \cdot y\right)\right) - \left(\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right) + x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right) + x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right)}{2 \cdot \left(y \cdot y\right) - \left(x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right) + \color{blue}{\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right)}\right)} \]

   4. sub0-neg N/A

      \[\leadsto \frac{\left(2 \cdot \left(y \cdot y\right)\right) \cdot \left(2 \cdot \left(y \cdot y\right)\right) - \left(\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right) + x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right) + x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right)}{2 \cdot \left(y \cdot y\right) - \left(x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right) + \left(\mathsf{neg}\left(y \cdot y\right)\right) \cdot \left(\color{blue}{y} \cdot y\right)\right)} \]

   5. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\left(2 \cdot \left(y \cdot y\right)\right) \cdot \left(2 \cdot \left(y \cdot y\right)\right) - \left(\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right) + x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right) + x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right)}{2 \cdot \left(y \cdot y\right) - \left(x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right) - \color{blue}{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{\left(2 \cdot \left(y \cdot y\right)\right) \cdot \left(2 \cdot \left(y \cdot y\right)\right) - \left(\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right) + x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(0 - y \cdot y\right) \cdot \left(y \cdot y\right) + x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right)\right)}{2 \cdot \left(y \cdot y\right) - \left(x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right) - y \cdot \color{blue}{\left(y \cdot \left(y \cdot y\right)\right)}\right)} \]

6. Applied egg-rr 18.8%

   \[\leadsto \color{blue}{\frac{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot 4 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 9\right)\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 9\right)\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}{y \cdot \left(y \cdot 2\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 9\right)\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}} \]

7. Final simplification 18.8%

   \[\leadsto \frac{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot 4 - \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 9\right)\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 9\right)\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)}{y \cdot \left(2 \cdot y\right) + \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right) - \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 9\right)\right)\right)} \]
8. Add Preprocessing

Alternative 3: 18.8% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 9\right)\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(y \cdot y\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- (* x (* x (* (* x x) 9.0))) (* y (* y (* y y)))) (* (* y y) -2.0)))

C

double code(double x, double y) {
	return ((x * (x * ((x * x) * 9.0))) - (y * (y * (y * y)))) - ((y * y) * -2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (x * ((x * x) * 9.0d0))) - (y * (y * (y * y)))) - ((y * y) * (-2.0d0))
end function

Java

public static double code(double x, double y) {
	return ((x * (x * ((x * x) * 9.0))) - (y * (y * (y * y)))) - ((y * y) * -2.0);
}

Python

def code(x, y):
	return ((x * (x * ((x * x) * 9.0))) - (y * (y * (y * y)))) - ((y * y) * -2.0)

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(x * Float64(Float64(x * x) * 9.0))) - Float64(y * Float64(y * Float64(y * y)))) - Float64(Float64(y * y) * -2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (x * ((x * x) * 9.0))) - (y * (y * (y * y)))) - ((y * y) * -2.0);
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y * y), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-+l- N/A

      \[\leadsto 9 \cdot {x}^{4} - \color{blue}{\left({y}^{4} - 2 \cdot \left(y \cdot y\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto 9 \cdot {x}^{4} - \left({y}^{4} + \color{blue}{\left(\mathsf{neg}\left(2 \cdot \left(y \cdot y\right)\right)\right)}\right) \]

   3. associate--r+ N/A

      \[\leadsto \left(9 \cdot {x}^{4} - {y}^{4}\right) - \color{blue}{\left(\mathsf{neg}\left(2 \cdot \left(y \cdot y\right)\right)\right)} \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(9 \cdot {x}^{4} - {y}^{4}\right), \color{blue}{\left(\mathsf{neg}\left(2 \cdot \left(y \cdot y\right)\right)\right)}\right) \]

4. Applied egg-rr 18.8%

   \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(9 \cdot \left(x \cdot x\right)\right)\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(y \cdot y\right) \cdot -2} \]

5. Final simplification 18.8%

   \[\leadsto \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot 9\right)\right) - y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(y \cdot y\right) \cdot -2 \]
6. Add Preprocessing

Alternative 4: 11.1% accurate, 42.6× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 2.0 (* y y)))

C

double code(double x, double y) {
	return 2.0 * (y * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 * (y * y)
end function

Java

public static double code(double x, double y) {
	return 2.0 * (y * y);
}

Python

def code(x, y):
	return 2.0 * (y * y)

Julia

function code(x, y)
	return Float64(2.0 * Float64(y * y))
end

MATLAB

function tmp = code(x, y)
	tmp = 2.0 * (y * y);
end

Wolfram

code[x_, y_] := N[(2.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.8%

   \[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2 \cdot {y}^{2} - {y}^{4}} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto 2 \cdot {y}^{2} - {y}^{\left(2 \cdot \color{blue}{2}\right)} \]

   2. pow-sqr N/A

      \[\leadsto 2 \cdot {y}^{2} - {y}^{2} \cdot \color{blue}{{y}^{2}} \]

   3. distribute-rgt-out-- N/A

      \[\leadsto {y}^{2} \cdot \color{blue}{\left(2 - {y}^{2}\right)} \]

   4. unsub-neg N/A

      \[\leadsto {y}^{2} \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left({y}^{2}\right)\right)}\right) \]

   5. mul-1-neg N/A

      \[\leadsto {y}^{2} \cdot \left(2 + -1 \cdot \color{blue}{{y}^{2}}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(2 + -1 \cdot {y}^{2}\right)}\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{2} + -1 \cdot {y}^{2}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{2} + -1 \cdot {y}^{2}\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(2 + \left(\mathsf{neg}\left({y}^{2}\right)\right)\right)\right) \]

   10. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(2 - \color{blue}{{y}^{2}}\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(2, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(2, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

   13. *-lowering-*.f64 1.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

5. Simplified 1.5%

   \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(2 - y \cdot y\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \color{blue}{2}\right) \]
7. Step-by-step derivation

8. Simplified 11.1%

   \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{2} \]

9. Final simplification 11.1%

   \[\leadsto 2 \cdot \left(y \cdot y\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024162 
(FPCore (x y)
  :name "From Rump in a 1983 paper"
  :precision binary64
  :pre (and (== x 10864.0) (== y 18817.0))
  (+ (- (* 9.0 (pow x 4.0)) (pow y 4.0)) (* 2.0 (* y y))))